Finite Group Databasebeta

← #27279127

Subgroups — #27279127

Order 13568.

572 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277318C211yesmaximal, characteristic
2#20274805C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272594C212yesmaximal
5C2×C53×#400C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810509—24no—
8#16810486—21no—
9#16810483C2212yes—
10C4×C22⋊Dic53—21no—
11#16810385—24no—
12#16810349—22no—
13C22×C4⋊C4×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400—22no—
16C53×#400C414yescomplemented
17C2×#95×C53—21no—
18C2×#95×C53—21no—
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21C2×C4⋊Dic53D414yes—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—22no—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133124—22no—
27#13133124—24no—
28#13133124—22no—
29C23×C4×C53—21no—
30C23×C4×C53—21no—
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C4⋊C4×C53D412yes—
42#95×C53—24no—
43#95×C53—24no—
44#95×C53—21no—
45#95×C53C2×C412yes—
46C24×C53C2×D411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—44no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53#4112yes—
69C22×Dic53—24no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—42no—
74C22⋊Dic53—42no—
75C22⋊Dic53—42no—
76C4⋊Dic53—44no—
77C4×Dic53—82no—
78C2×#88322—44no—
79C2×#88322—42no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53C2×C22⋊C411yescharacteristic
83C23×C53#12011yescharacteristic
84C23×C53#12311yescharacteristic
85C23×C53#13512yes—
86C23×C53#13712yes—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—44no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—81no—
105C2×C4×C53—84no—
106C2×C4×C53—82no—
107C2×C4×C53—82no—
108C2×C4×C53C4×D412yes—
109C2×C4×C53#12112yes—
110C2×Dic53—44no—
111C2×Dic53—44no—
112C2×Dic53—44no—
113#88322—82no—
114#9022—531nomaximal, Sylow
115C22×C53—24no—
116C22×C53—24no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#38611yescharacteristic
122C22×C53#41611yescharacteristic
123C22×C53#41911yescharacteristic
124C22×C53#45212yes—
125C22×C53#45412yes—
126C22×C53#45812yes—
127C22×C53#48512yes—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—42no—
132C4×C53—82no—
133C4×C53—82no—
134Dic53—84no—
135C2×#400D5311yescharacteristic
136#2313—531no—
137#1824—532no—
138C2×#324—531no—
139#1294—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—22no—
143C2×C53#161514yes—
144C2×C53#187311yescharacteristic
145C2×C53#188811yescharacteristic
146C2×C53#189011yescharacteristic
147C22×C4⋊C4C2×D5311yescharacteristic
148C22×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C8—532no—
150#400—22no—
151#400Dic5314yescomplemented
152#387—1064no—
153#381—1061no—
154#380—532no—
155C4×C22⋊C4—1061no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95C2×D5311yescharacteristic
159#340—1064no—
160#324—1062no—
161C53#902211yescharacteristic, complemented, Sylow
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22⋊D5311yescharacteristic
165C23×C4C22⋊D5311yescharacteristic
166C23×C4C22×D5311yescharacteristic
167C22×C8—1062no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—534no—
174C2×C4⋊C4C2×Dic5312yes—
175C2×C4⋊C4C2×Dic5312yes—
176C2×C4⋊C4C22⋊D5312yes—
177C2×C22⋊C4—1062no—
178C2×C42—1062no—
179C22⋊C8—1062no—
180C22⋊C8—1062no—
181#95—21no—
182#95—24no—
183#95—24no—
184#95—1062no—
185#95—1064no—
186#95—1062no—
187#95C2×Dic5312yes—
188C24C2×C22⋊D5311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—1064no—
206C22×C4C22⋊Dic5312yes—
207C22×C4C22⋊Dic5312yes—
208C22×C4D4×D5312yes—
209C22×C4#1171994712yes—
210C22×C4C22×Dic5311yescharacteristic
211C22×C4C2×C22⋊D5311yescharacteristic
212C2×C8—2124no—
213C2×C8—2122no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—42no—
218C4⋊C4—2124no—
219C22⋊C4—2122no—
220C22⋊C4—2122no—
221C42—4242no—
222C23—22no—
223C23—22no—
224C23C2×C22⋊Dic5311yescharacteristic
225C23#1313318511yescharacteristic
226C23#1313319811yescharacteristic
227C23#1313323712yes—
228C23#1313324812yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—2124no—
251C2×C4—2124no—
252C2×C4—2124no—
253C2×C4D4×Dic5312yes—
254C2×C4#1313318912yes—
255C8—4242no—
256C22—24no—
257C22—24no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22#1681050311yescharacteristic
263C22#1681061911yescharacteristic
264C22#1681063211yescharacteristic
265C22#1681078312yes—
266C22#1681079512yes—
267C22#1681081212yes—
268C22#1681096112yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—4244no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2#2027374414yes—
280C2#2027502811yescharacteristic
281C2#2027511911yescharacteristic
282C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#400C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#400D53
hypocenterC53#9022
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
16. C53×#400C4semidirect
151. #400Dic53semidirect
161. C53#9022semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202773182
#202748052
C2×#168103492
#202725942
C2×C53×#4002
#902253